Add the chicken and cook to hot oil, turning once or twice and adjusting the heat to prevent burning, until brown. Transfer to a plate and tent with foil to keep warm.

Add shallot to the pan and cook, stirring, until fragrant, about 30 seconds. Add mushrooms; cook, stirring occasionally, until tender, about 2 minutes. Pour in vermouth (or wine); simmer until almost evaporated, scraping up any browned bits, about 1 minute. Pour in broth and cook until reduced by half, 1 to 2 minutes. Stir in cream and chives (or scallions); return to a simmer. Return the chicken to the pan, turn to coat with sauce and cook until heated through, about 1 minute.